Mathematical Journeys: A Tale of Two Contexts

Sometimes the same procedure shows up in two different contexts. This is especially common in the fields of math and science, as science employs in real-world application many of the techniques we learn in their abstract form in math class. For some reason, the principle as shown in a high-school science class is often much harder for students to understand than it was in the math class. (My personal theory is that science teachers are applying the concept in a way that changes how they explain how it works, and they probably have not collaborated with the student's math teacher to ensure they're reinforcing the same terminology.) Last week one of my students ran into this phenomenon in her own work; a concept from last year's math class showed up in her physics class. To help her understand it, we went back to the original math concept and talked about proportions.

The science homework she was struggling with was the old chestnut about unit conversions; rows and rows of fractions set up in perplexing fashion, with equals signs between each one and confusing scribbles across and between them. When taught in physics, the idea of unit conversions frequently becomes clouded and nonsensical – it becomes a feat of rote memorization to attempt to figure out what the correct procedure is for each specific case. In truth, unit conversions are a simple case of mathematical proportions, and the easiest way for my student to understand it was to connect it to the original context in which she learned about proportions last year – similar triangles.

A proportion is a pair of fractions that are set equal to each other. Remember – one of the definitions of a fraction is a ratio (comparison of two quantities), so setting two fractions equal to each other signifies that these two ratios are equivalent. We use this to understand similar triangles in early algebra, working through the logical reasoning process to grasp the concept that increasing all parts of the figure by the same amount will maintain the relationship of parts to each other. Unit conversions are simply another application of this same idea.

Let's say your physics problem involves the length of a bridge. The problem states that the bridge is 3200 feet long, and wants to know how many meters long that is. We're not doing anything to change the bridge itself, we're simply changing how we measure it. That can be analogous to finding an equivalent fraction – we're not changing the value of the fraction, just how it's written. No matter whether we measure the bridge in feet, meters, inches, or miles, the relationship of the various parts of the bridge to each other (the proportions of the bridge) will stay the same. So this is a proportion problem.

For a proportion problem to work, we need two fractions. We only have one number in the problem, so where are the rest of the puzzle pieces? They're implied, in the form of something called a conversion ratio. The conversion ratio is the fraction which indicates how the two units relate to each other – in our case, how many feet are in one meter. If we know that, it's a simple similar-triangles-type problem to figure out our conversion.

So what's the conversion ratio for feet into meters? Well, usually in a problem like this, the conversion ratios are either easy to find out for yourself, or they're provided for you. A quick internet search reveals that one meter is equal to 3.28084 feet. So, without doing any math just yet, common sense tells us that converting the 3200 feet of the bridge into meters should give us a smaller number than we started with – there are multiple feet in one meter, so we'll be dividing the number of feet up into one-meter chunks. (I find this kind of common sense flagging is helpful for these kinds of problems.)

So let's figure out our proportion!

1 meter/3.28084 feet = x meters/3200 feet

I find it helpful to read this sentence to yourself as an analogy: “One meter is to 3.28084 feet as x meters are to 3200 feet.”

We know we have the proportion set up correctly because the matching units are on corresponding sides of each fraction. Just as a similar triangle problem needs the corresponding sides to match up with either the top or bottom of each fraction, unit conversions need to match up the units the same way.

You know what to do from here, right? Cross-multiply!

3.28084x = 3200(1)

x = 3200/3.28084

Hold up there for just a second. Remember a bit further up when I flagged for us that we'd be “ dividing the number of feet up into one-meter chunks”? Check it out – that's exactly what we're doing! Taking the total number of feet and dividing it by the number of feet per meter. That's another way it's sometimes described in science classes:

total feet [divided by] feet/meter

and when you divide by a fraction, you multiply by the reciprocal, so:

total feet * meter/feet

Sometimes science teachers will make a big show out of placing the fractions like this, so that they can physically cancel out the feet and all they have left is meters. I've found this to be helpful in calculations that require multiple conversions (inches to kilometers, for example), since it makes sure you keep the units straight, but it's just as easy to do the proportion problems one at a time, and these days you can usually find the conversion ratio to go straight from one to the other anyway. A quick internet search tells me the conversion ratio for the example I just gave is 39370.1 inches per kilometer. Boom. Done. Write your proportion and go to town.

Anyway, heading back to our original problem, we see that:

x = 3200 feet/3.28084 feet per meter

x = 975.36 meters

Which does satisfy our initial common sense check, as it's considerably smaller than the starting value.

In these examples, I find it much more useful to think of them as proportion problems in the vein of the similar triangle problems we practice so much in algebra. The algorithm is the same, it's just the context that differs, and if that's the case, why not think of the problem using whatever context makes the most sense to you?
For more on similar triangles, check out my blog post about that time my TV broke.
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